Production of Gravitational Waves in the Early Universe From turbulence triggered by first-order phase transitions

This paper investigates the production of primordial gravitational waves from first-order phase transitions in the early universe by modeling the resulting plasma turbulence with relativistic hydrodynamics and comparing two existing models against a new approach that incorporates freely decaying turbulence with a time-dependent de-coherence function.

The Gist

Here is an explanation of Yashmitha Kumaran's Master's thesis, translated from complex physics into a story you can visualize.

The Big Picture: Listening to the Baby Universe

Imagine the universe as a giant, expanding balloon. When it was just a baby (a tiny fraction of a second after the Big Bang), it was incredibly hot and dense. As it cooled down, it didn't just get colder; it underwent phase transitions, much like water turning into ice.

But this wasn't a smooth freeze. It was a violent, chaotic event. Think of it like a pot of water boiling. Bubbles of steam form, grow, and crash into each other. In the early universe, these "bubbles" were pockets of a new state of reality expanding through the old state.

When these bubbles collided, they didn't just make a sound; they created ripples in the fabric of space and time itself. These ripples are called Gravitational Waves.

This project is a detective story. The author, Yashmitha, is trying to figure out exactly what these ripples would look like today, billions of years later, so that future telescopes can "hear" them.

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The Problem: Modeling the Chaos

To predict what these waves look like, you have to model the "soup" of the early universe. It wasn't just static; it was a turbulent fluid.

Imagine stirring a cup of coffee. You create big swirls, which break into smaller swirls, which break into tiny swirls, until the energy disappears as heat. This is called turbulence.

The author looked at two existing "recipes" (models) that scientists had already written to describe this cosmic turbulence:

1. The "Stationary" Model (Model 1): This recipe assumes the turbulence is like a steady, humming machine. It uses a specific mathematical rule (the Kraichnan function) to guess how long the swirls stay connected before they break apart.

   • The Flaw: It works well for slow, gentle flows (low Reynolds number), but the early universe was moving at supersonic speeds. This recipe breaks down in extreme chaos.

2. The "Top Hat" Model (Model 2): This recipe assumes the turbulence is a short, sharp burst (like a hammer hit) that stops abruptly. It uses a "Top Hat" shape to decide when the swirls stop talking to each other.

   • The Flaw: It's a bit too rigid. It ignores how the fluid actually moves and sweeps things along.

The Solution: The "Sweeping" Model

Yashmitha's goal was to build a new, better recipe that combines the best parts of the other two while fixing their mistakes.

She introduced a concept called "Sweeping Decorrelation."

The Analogy:
Imagine you are standing on a sidewalk watching leaves swirl in a puddle.

• The Old Way: You assume the leaves swirl in place and slowly lose their pattern.
• The New Way (Sweeping): You realize the wind is blowing the entire puddle past you. The leaves aren't just swirling; they are being swept away by the current.

In the early universe, the "wind" is the massive speed of the plasma (the hot fluid). The author realized that the turbulence isn't just decaying on its own; it's being swept away by the fast-moving fluid. By adding this "sweeping" effect into the math, the model becomes much more accurate for the high-speed environment of the early universe.

What Did She Find?

Yashmitha took the math from the two old models, mixed them with her new "sweeping" idea, and ran the numbers.

1. The Frequency: She calculated what "pitch" (frequency) these gravitational waves would have.
2. The Volume: She calculated how loud (amplitude) they would be.
3. The Shape: She found that her new model creates a unique "signature" or shape in the data that is different from the old models.

The Result: Her new model predicts that the gravitational waves from the early universe might be slightly different than we thought before. Specifically, it handles the extreme speeds (high Reynolds numbers) of the early universe much better than the old recipes.

Why Does This Matter?

The "Fingerprint" of Creation:
Gravitational waves are like ghosts. They pass through everything—stars, gas, dust—without stopping. If we can detect the gravitational waves from these ancient bubble collisions, we get a direct "fingerprint" of the universe when it was less than a second old.

The Future Hunt:
Currently, we can't hear these specific waves. But future space telescopes (like LISA or DECIGO) are being built to listen for them.

Yashmitha's work is like tuning the radio. By refining the mathematical models of what the signal should look like, she helps the scientists building those telescopes know exactly what frequency to tune into. If they find a signal that matches her "Sweeping Model," it could prove that the universe went through these violent phase transitions and help us understand the fundamental forces of nature.

Summary in One Sentence

This paper builds a better mathematical map of the chaotic "storm" in the baby universe, helping us predict exactly what the echoes of that storm (gravitational waves) should sound like, so we can finally hear the birth of our universe.

Technical Summary

Here is a detailed technical summary of the Master's thesis "Production of Gravitational Waves in the Early Universe: From turbulence triggered by first-order phase transitions" by Yashmitha Kumaran (University of Sussex, 2018).

1. Problem Statement

The project investigates the generation of a stochastic background of primordial gravitational waves (GWs) resulting from first-order phase transitions in the early universe.

• Context: During the early universe's cooling, symmetry breaking occurred via first-order phase transitions, leading to the nucleation, growth, and collision of vacuum bubbles.
• Mechanism: These bubble collisions and the subsequent fluid dynamics (shock waves and rarefaction waves) generate anisotropic stresses in the cosmic plasma. These stresses act as a source for gravitational waves.
• The Gap: While the existence of these waves is theorized, accurately modeling the turbulence generated by these transitions is complex. Existing models often rely on simplifying assumptions regarding the temporal decorrelation of velocity fields and the nature of the turbulence (stationary vs. decaying), particularly at high Reynolds numbers typical of the early universe. The goal is to refine these models to better predict the GW energy density, amplitude, and frequency spectra for future detection.

2. Methodology

The author employs relativistic hydrodynamics and Kolmogorov turbulence theory to model the plasma flow. The study involves three distinct modeling approaches:

A. Theoretical Framework

• Hydrodynamics: The plasma is treated as a relativistic perfect fluid. The source of GWs is the transverse-traceless part of the energy-momentum tensor ($\Pi_{ij}$), derived from fluid velocity correlations.
• Turbulence: The study assumes the turbulence follows Kolmogorov scaling ($E(k) \propto k^{-5/3}$), where energy cascades from large eddies to small dissipative scales.
• Source Duration: The phase transition is modeled as a "short-lasting source" (active only during the bubble percolation phase), neglecting the expansion of the universe during this brief interval.

B. The Three Models Compared

The thesis reproduces two existing reference models and introduces a new hybrid model:

1. Model 1: Stationary Turbulence (Reference: Gogoberidze et al., 2007)

   • Assumption: Turbulence is stationary and homogeneous.
   • Decorrelation: Uses the Kraichnan exponential function for temporal decorrelation: $f(\eta_k, \tau) = \exp(-\frac{\pi}{4}\eta_k^2 \tau^2)$.
   • Limitation: This function is theoretically valid primarily for low Reynolds numbers and assumes a specific time-dependence that may not hold for high-energy early universe conditions.

2. Model 2: Top Hat Correlation (Reference: Caprini et al., 2009)

   • Assumption: Focuses on a "short-lasting source" where turbulence decays freely after the phase transition.
   • Decorrelation: Uses a Top Hat approximation (Heaviside step function) for the correlation of unequal times. It assumes the source is correlated only within a specific time window ($t_c/k$) and uncorrelated outside it.
   • Limitation: While it accounts for the finite duration of the source, it simplifies the velocity power spectrum correlation significantly.

3. Model 3: The New "Sweeping Decorrelation" Model (Proposed)

   • Innovation: This model attempts to bridge the gap between the two references.
   • Mechanism: It adopts the Sweeping Hypothesis (Kraichnan) to resolve issues with high Reynolds numbers. It assumes spacetime correlations are dominated by the "sweeping" of eddies by the root-mean-square (RMS) velocity of the plasma ($\bar{U}$).
   • Implementation:
     • It retains the Top Hat approximation structure for the source duration (from Model 2).
     • It replaces the standard decorrelation function with a time-dependent de-coherence function derived from the sweeping hypothesis: $f(\eta_k, t_-) = \exp(-\frac{1}{2}\bar{U}^2 \eta_k^2 t_-^2)$.
     • It calculates the anisotropic stress ($\Pi$) using both transverse and longitudinal velocity components, unlike some simplified models that only consider transverse modes.

3. Key Contributions

• Reproduction of Reference Models: The author successfully reproduced the gravitational wave power spectra for both the Stationary Turbulence model and the Top Hat model, validating the computational framework.
• Hybrid Model Development: The primary contribution is the formulation of a new model that combines the Top Hat correlation (for source duration) with the Sweeping decorrelation function (for high Reynolds number validity).
• Analytical Derivation: The thesis provides detailed algebraic derivations for the anisotropic stress tensor ($\Pi$) by splitting it into transverse ($\perp$) and longitudinal ($\parallel$) components and integrating over wavenumbers.
• Spectral Analysis: The author computed the rate of spectral density, the GW power spectrum, and the characteristic strain amplitude ($h_c$) for all three models.

4. Results

The analysis yielded the following spectral behaviors (plotted against non-dimensionalized wavenumber $z$):

• Anisotropic Stress: The new model shows a power-law variation of $k^{-17/6}$ before integration and $k^{-11/3}$ after integration, consistent with the reference models but with modified amplitude due to the sweeping effect.
• Power Spectrum ($P_{GW}$):
  • The new model exhibits a $k^3$ rise at low frequencies.
  • It peaks at $z \approx 3.56$ (corresponding to a specific frequency).
  • Post-peak, it decays as $k^{-5/3}$, matching the Kolmogorov spectrum.
• Amplitude vs. Frequency ($h_c$):
  • The amplitude scales as $\sqrt{k}$ in the rising phase.
  • It peaks at $h_c \approx 8.3 \times 10^{-21}$ (for Mach number $M=1$).
  • It decays as $k^{-11/6}$ in the high-frequency tail.
• Comparison: The new model (Green in plots) successfully retains the spectral shape of the reference models (Blue/Red) but modifies the peak and decay rates to account for the sweeping effect, effectively breaking the limitation of the stationary model at high Reynolds numbers.

5. Significance and Limitations

Significance:

• Detection Prospects: The refined spectral predictions are crucial for future space-based interferometers like LISA, DECIGO, and SKA. Accurate modeling of the peak frequency and amplitude determines whether these instruments can detect the stochastic background from the early universe.
• Physics Validation: Detecting these waves would provide direct evidence of first-order phase transitions, potentially verifying theories of symmetry breaking (e.g., the Higgs mechanism) and high-energy particle physics beyond the Standard Model.
• Methodological Advancement: The "Sweeping Decorrelation" approach offers a more physically robust treatment of turbulence in the early universe, particularly addressing the high Reynolds number regime where standard stationary assumptions fail.

Limitations:

• Free Decay: The proposed model does not fully account for freely decaying turbulence (where the RMS velocity decreases over time). It assumes a constant RMS velocity during the source duration.
• Source Duration Assumption: The model assumes the turbulence source is short-lived (much less than a Hubble time) and ignores the expansion of the universe during the phase transition. While this simplifies the math, it may be an oversimplification for certain cosmological scenarios.
• Pedagogical Nature: As noted in the author's preface, this work serves as a pedagogical study and a foundation for later, more comprehensive research (citing Hindmarsh et al., 2019–2021) that further developed these frameworks.

Conclusion

The thesis successfully demonstrates that combining the Top Hat correlation (for source duration) with the Sweeping decorrelation hypothesis (for high Reynolds number turbulence) yields a more robust model for predicting primordial gravitational waves. The resulting spectra provide specific targets for future observational campaigns, enhancing the possibility of detecting the "fingerprint" of the early universe's violent phase transitions.